ON REFRACTION
In the same way as the effects of Reflexion have been explained by
waves of light reflected at the surface of polished bodies, we will
explain transparency and the phenomena of refraction by waves which
spread within and across diaphanous bodies, both solids, such as
glass, and liquids, such as water, oils, etc. But in order that it may
not seem strange to suppose this passage of waves in the interior of
these bodies, I will first show that one may conceive it possible in
more than one mode.
First, then, if the ethereal matter cannot penetrate transparent
bodies at all, their own particles would be able to communicate
successively the movement of the waves, the same as do those of the
Ether, supposing that, like those, they are of a nature to act as a
spring. And this is easy to conceive as regards water and other
transparent liquids, they being composed of detached particles. But it
may seem more difficult as regards glass and other transparent and
hard bodies, because their solidity does not seem to permit them to
receive movement except in their whole mass at the same time. This,
however, is not necessary because this solidity is not such as it
appears to us, it being probable rather that these bodies are composed
of particles merely placed close to one another and held together by
some pressure from without of some other matter, and by the
irregularity of their shapes. For primarily their rarity is shown by
the facility with which there passes through them the matter of the
vortices of the magnet, and that which causes gravity. Further, one
cannot say that these bodies are of a texture similar to that of a
sponge or of light bread, because the heat of the fire makes them flow
and thereby changes the situation of the particles amongst themselves.
It remains then that they are, as has been said, assemblages of
particles which touch one another without constituting a continuous
solid. This being so, the movement which these particles receive to
carry on the waves of light, being merely communicated from some of
them to others, without their going for that purpose out of their
places or without derangement, it may very well produce its effect
without prejudicing in any way the apparent solidity of the compound.
By pressure from without, of which I have spoken, must not be
understood that of the air, which would not be sufficient, but that of
some other more subtle matter, a pressure which I chanced upon by
experiment long ago, namely in the case of water freed from air, which
remains suspended in a tube open at its lower end, notwithstanding
that the air has been removed from the vessel in which this tube is
enclosed.
One can then in this way conceive of transparency in a solid without
any necessity that the ethereal matter which serves for light should
pass through it, or that it should find pores in which to insinuate
itself. But the truth is that this matter not only passes through
solids, but does so even with great facility; of which the experiment
of Torricelli, above cited, is already a proof. Because on the
quicksilver and the water quitting the upper part of the glass tube,
it appears that it is immediately filled with ethereal matter, since
light passes across it. But here is another argument which proves this
ready penetrability, not only in transparent bodies but also in all
others.
When light passes across a hollow sphere of glass, closed on all
sides, it is certain that it is full of ethereal matter, as much as
the spaces outside the sphere. And this ethereal matter, as has been
shown above, consists of particles which just touch one another. If
then it were enclosed in the sphere in such a way that it could not
get out through the pores of the glass, it would be obliged to follow
the movement of the sphere when one changes its place: and it would
require consequently almost the same force to impress a certain
velocity on this sphere, when placed on a horizontal plane, as if it
were full of water or perhaps of quicksilver: because every body
resists the velocity of the motion which one would give to it, in
proportion to the quantity of matter which it contains, and which is
obliged to follow this motion. But on the contrary one finds that the
sphere resists the impress of movement only in proportion to the
quantity of matter of the glass of which it is made. Then it must be
that the ethereal matter which is inside is not shut up, but flows
through it with very great freedom. We shall demonstrate hereafter
that by this process the same penetrability may be inferred also as
relating to opaque bodies.
The second mode then of explaining transparency, and one which appears
more probably true, is by saying that the waves of light are carried
on in the ethereal matter, which continuously occupies the interstices
or pores of transparent bodies. For since it passes through them
continuously and freely, it follows that they are always full of it.
And one may even show that these interstices occupy much more space
than the coherent particles which constitute the bodies. For if what
we have just said is true: that force is required to impress a certain
horizontal velocity on bodies in proportion as they contain coherent
matter; and if the proportion of this force follows the law of
weights, as is confirmed by experiment, then the quantity of the
constituent matter of bodies also follows the proportion of their
weights. Now we see that water weighs only one fourteenth part as much
as an equal portion of quicksilver: therefore the matter of the water
does not occupy the fourteenth part of the space which its mass
obtains. It must even occupy much less of it, since quicksilver is
less heavy than gold, and the matter of gold is by no means dense, as
follows from the fact that the matter of the vortices of the magnet
and of that which is the cause of gravity pass very freely through it.
But it may be objected here that if water is a body of so great
rarity, and if its particles occupy so small a portion of the space of
its apparent bulk, it is very strange how it yet resists Compression
so strongly without permitting itself to be condensed by any force
which one has hitherto essayed to employ, preserving even its entire
liquidity while subjected to this pressure.
This is no small difficulty. It may, however, be resolved by saying
that the very violent and rapid motion of the subtle matter which
renders water liquid, by agitating the particles of which it is
composed, maintains this liquidity in spite of the pressure which
hitherto any one has been minded to apply to it.
The rarity of transparent bodies being then such as we have said, one
easily conceives that the waves might be carried on in the ethereal
matter which fills the interstices of the particles. And, moreover,
one may believe that the progression of these waves ought to be a
little slower in the interior of bodies, by reason of the small
detours which the same particles cause. In which different velocity of
light I shall show the cause of refraction to consist.
Before doing so, I will indicate the third and last mode in which
transparency may be conceived; which is by supposing that the movement
of the waves of light is transmitted indifferently both in the
particles of the ethereal matter which occupy the interstices of
bodies, and in the particles which compose them, so that the movement
passes from one to the other. And it will be seen hereafter that this
hypothesis serves excellently to explain the double refraction of
certain transparent bodies.
Should it be objected that if the particles of the ether are smaller
than those of transparent bodies (since they pass through their
intervals), it would follow that they can communicate to them but
little of their movement, it may be replied that the particles of
these bodies are in turn composed of still smaller particles, and so
it will be these secondary particles which will receive the movement
from those of the ether.
Furthermore, if the particles of transparent bodies have a recoil a
little less prompt than that of the ethereal particles, which nothing
hinders us from supposing, it will again follow that the progression
of the waves of light will be slower in the interior of such bodies
than it is outside in the ethereal matter.
All this I have found as most probable for the mode in which the waves
of light pass across transparent bodies. To which it must further be
added in what respect these bodies differ from those which are opaque;
and the more so since it might seem because of the easy penetration of
bodies by the ethereal matter, of which mention has been made, that
there would not be any body that was not transparent. For by the same
reasoning about the hollow sphere which I have employed to prove the
smallness of the density of glass and its easy penetrability by the
ethereal matter, one might also prove that the same penetrability
obtains for metals and for every other sort of body. For this sphere
being for example of silver, it is certain that it contains some of
the ethereal matter which serves for light, since this was there as
well as in the air when the opening of the sphere was closed. Yet,
being closed and placed upon a horizontal plane, it resists the
movement which one wishes to give to it, merely according to the
quantity of silver of which it is made; so that one must conclude, as
above, that the ethereal matter which is enclosed does not follow the
movement of the sphere; and that therefore silver, as well as glass,
is very easily penetrated by this matter. Some of it is therefore
present continuously and in quantities between the particles of silver
and of all other opaque bodies: and since it serves for the
propagation of light it would seem that these bodies ought also to be
transparent, which however is not the case.
Whence then, one will say, does their opacity come? Is it because the
particles which compose them are soft; that is to say, these particles
being composed of others that are smaller, are they capable of
changing their figure on receiving the pressure of the ethereal
particles, the motion of which they thereby damp, and so hinder the
continuance of the waves of light? That cannot be: for if the
particles of the metals are soft, how is it that polished silver and
mercury reflect light so strongly? What I find to be most probable
herein, is to say that metallic bodies, which are almost the only
really opaque ones, have mixed amongst their hard particles some soft
ones; so that some serve to cause reflexion and the others to hinder
transparency; while, on the other hand, transparent bodies contain
only hard particles which have the faculty of recoil, and serve
together with those of the ethereal matter for the propagation of the
waves of light, as has been said.
[Illustration]
Let us pass now to the explanation of the effects of Refraction,
assuming, as we have done, the passage of waves of light through
transparent bodies, and the diminution of velocity which these same
waves suffer in them.
The chief property of Refraction is that a ray of light, such as AB,
being in the air, and falling obliquely upon the polished surface of a
transparent body, such as FG, is broken at the point of incidence B,
in such a way that with the straight line DBE which cuts the surface
perpendicularly it makes an angle CBE less than ABD which it made with
the same perpendicular when in the air. And the measure of these
angles is found by describing, about the point B, a circle which cuts
the radii AB, BC. For the perpendiculars AD, CE, let fall from the
points of intersection upon the straight line DE, which are called the
Sines of the angles ABD, CBE, have a certain ratio between themselves;
which ratio is always the same for all inclinations of the incident
ray, at least for a given transparent body. This ratio is, in glass,
very nearly as 3 to 2; and in water very nearly as 4 to 3; and is
likewise different in other diaphanous bodies.
Another property, similar to this, is that the refractions are
reciprocal between the rays entering into a transparent body and those
which are leaving it. That is to say that if the ray AB in entering
the transparent body is refracted into BC, then likewise CB being
taken as a ray in the interior of this body will be refracted, on
passing out, into BA.
[Illustration]
To explain then the reasons of these phenomena according to our
principles, let AB be the straight line which represents a plane
surface bounding the transparent substances which lie towards C and
towards N. When I say plane, that does not signify a perfect evenness,
but such as has been understood in treating of reflexion, and for the
same reason. Let the line AC represent a portion of a wave of light,
the centre of which is supposed so distant that this portion may be
considered as a straight line. The piece C, then, of the wave AC, will
in a certain space of time have advanced as far as the plane AB
following the straight line CB, which may be imagined as coming from
the luminous centre, and which consequently will cut AC at right
angles. Now in the same time the piece A would have come to G along
the straight line AG, equal and parallel to CB; and all the portion of
wave AC would be at GB if the matter of the transparent body
transmitted the movement of the wave as quickly as the matter of the
Ether. But let us suppose that it transmits this movement less
quickly, by one-third, for instance. Movement will then be spread from
the point A, in the matter of the transparent body through a distance
equal to two-thirds of CB, making its own particular spherical wave
according to what has been said before. This wave is then represented
by the circumference SNR, the centre of which is A, and its
semi-diameter equal to two-thirds of CB. Then if one considers in
order the other pieces H of the wave AC, it appears that in the same
time that the piece C reaches B they will not only have arrived at the
surface AB along the straight lines HK parallel to CB, but that, in
addition, they will have generated in the diaphanous substance from
the centres K, partial waves, represented here by circumferences the
semi-diameters of which are equal to two-thirds of the lines KM, that
is to say, to two-thirds of the prolongations of HK down to the
straight line BG; for these semi-diameters would have been equal to
entire lengths of KM if the two transparent substances had been of the
same penetrability.
Now all these circumferences have for a common tangent the straight
line BN; namely the same line which is drawn as a tangent from the
point B to the circumference SNR which we considered first. For it is
easy to see that all the other circumferences will touch the same BN,
from B up to the point of contact N, which is the same point where AN
falls perpendicularly on BN.
It is then BN, which is formed by small arcs of these circumferences,
which terminates the movement that the wave AC has communicated within
the transparent body, and where this movement occurs in much greater
amount than anywhere else. And for that reason this line, in
accordance with what has been said more than once, is the propagation
of the wave AC at the moment when its piece C has reached B. For there
is no other line below the plane AB which is, like BN, a common
tangent to all these partial waves. And if one would know how the wave
AC has come progressively to BN, it is necessary only to draw in the
same figure the straight lines KO parallel to BN, and all the lines KL
parallel to AC. Thus one will see that the wave CA, from being a
straight line, has become broken in all the positions LKO
successively, and that it has again become a straight line at BN. This
being evident by what has already been demonstrated, there is no need
to explain it further.
Now, in the same figure, if one draws EAF, which cuts the plane AB at
right angles at the point A, since AD is perpendicular to the wave AC,
it will be DA which will mark the ray of incident light, and AN which
was perpendicular to BN, the refracted ray: since the rays are nothing
else than the straight lines along which the portions of the waves
advance.
Whence it is easy to recognize this chief property of refraction,
namely that the Sine of the angle DAE has always the same ratio to the
Sine of the angle NAF, whatever be the inclination of the ray DA: and
that this ratio is the same as that of the velocity of the waves in
the transparent substance which is towards AE to their velocity in the
transparent substance towards AF. For, considering AB as the radius of
a circle, the Sine of the angle BAC is BC, and the Sine of the angle
ABN is AN. But the angle BAC is equal to DAE, since each of them added
to CAE makes a right angle. And the angle ABN is equal to NAF, since
each of them with BAN makes a right angle. Then also the Sine of the
angle DAE is to the Sine of NAF as BC is to AN. But the ratio of BC to
AN was the same as that of the velocities of light in the substance
which is towards AE and in that which is towards AF; therefore also
the Sine of the angle DAE will be to the Sine of the angle NAF the
same as the said velocities of light.
To see, consequently, what the refraction will be when the waves of
light pass into a substance in which the movement travels more quickly
than in that from which they emerge (let us again assume the ratio of
3 to 2), it is only necessary to repeat all the same construction and
demonstration which we have just used, merely substituting everywhere
3/2 instead of 2/3. And it will be found by the same reasoning, in
this other figure, that when the piece C of the wave AC shall have
reached the surface AB at B, all the portions of the wave AC will
have advanced as far as BN, so that BC the perpendicular on AC is to
AN the perpendicular on BN as 2 to 3. And there will finally be this
same ratio of 2 to 3 between the Sine of the angle BAD and the Sine of
the angle FAN.
Hence one sees the reciprocal relation of the refractions of the ray
on entering and on leaving one and the same transparent body: namely
that if NA falling on the external surface AB is refracted into the
direction AD, so the ray AD will be refracted on leaving the
transparent body into the direction AN.
[Illustration]
One sees also the reason for a noteworthy accident which happens in
this refraction: which is this, that after a certain obliquity of the
incident ray DA, it begins to be quite unable to penetrate into the
other transparent substance. For if the angle DAQ or CBA is such that
in the triangle ACB, CB is equal to 2/3 of AB, or is greater, then AN
cannot form one side of the triangle ANB, since it becomes equal to or
greater than AB: so that the portion of wave BN cannot be found
anywhere, neither consequently can AN, which ought to be perpendicular
to it. And thus the incident ray DA does not then pierce the surface
AB.
When the ratio of the velocities of the waves is as two to three, as
in our example, which is that which obtains for glass and air, the
angle DAQ must be more than 48 degrees 11 minutes in order that the
ray DA may be able to pass by refraction. And when the ratio of the
velocities is as 3 to 4, as it is very nearly in water and air, this
angle DAQ must exceed 41 degrees 24 minutes. And this accords
perfectly with experiment.
But it might here be asked: since the meeting of the wave AC against
the surface AB ought to produce movement in the matter which is on the
other side, why does no light pass there? To which the reply is easy
if one remembers what has been said before. For although it generates
an infinitude of partial waves in the matter which is at the other
side of AB, these waves never have a common tangent line (either
straight or curved) at the same moment; and so there is no line
terminating the propagation of the wave AC beyond the plane AB, nor
any place where the movement is gathered together in sufficiently
great quantity to produce light. And one will easily see the truth of
this, namely that CB being larger than 2/3 of AB, the waves excited
beyond the plane AB will have no common tangent if about the centres K
one then draws circles having radii equal to 3/2 of the lengths LB to
which they correspond. For all these circles will be enclosed in one
another and will all pass beyond the point B.
Now it is to be remarked that from the moment when the angle DAQ is
smaller than is requisite to permit the refracted ray DA to pass into
the other transparent substance, one finds that the interior reflexion
which occurs at the surface AB is much augmented in brightness, as is
easy to realize by experiment with a triangular prism; and for this
our theory can afford this reason. When the angle DAQ is still large
enough to enable the ray DA to pass, it is evident that the light from
the portion AC of the wave is collected in a minimum space when it
reaches BN. It appears also that the wave BN becomes so much the
smaller as the angle CBA or DAQ is made less; until when the latter is
diminished to the limit indicated a little previously, this wave BN is
collected together always at one point. That is to say, that when the
piece C of the wave AC has then arrived at B, the wave BN which is the
propagation of AC is entirely reduced to the same point B. Similarly
when the piece H has reached K, the part AH is entirely reduced to the
same point K. This makes it evident that in proportion as the wave CA
comes to meet the surface AB, there occurs a great quantity of
movement along that surface; which movement ought also to spread
within the transparent body and ought to have much re-enforced the
partial waves which produce the interior reflexion against the surface
AB, according to the laws of reflexion previously explained.
And because a slight diminution of the angle of incidence DAQ causes
the wave BN, however great it was, to be reduced to zero, (for this
angle being 49 degrees 11 minutes in the glass, the angle BAN is still
11 degrees 21 minutes, and the same angle being reduced by one degree
only the angle BAN is reduced to zero, and so the wave BN reduced to a
point) thence it comes about that the interior reflexion from being
obscure becomes suddenly bright, so soon as the angle of incidence is
such that it no longer gives passage to the refraction.
Now as concerns ordinary external reflexion, that is to say which
occurs when the angle of incidence DAQ is still large enough to enable
the refracted ray to penetrate beyond the surface AB, this reflexion
should occur against the particles of the substance which touches the
transparent body on its outside. And it apparently occurs against the
particles of the air or others mingled with the ethereal particles and
larger than they. So on the other hand the external reflexion of these
bodies occurs against the particles which compose them, and which are
also larger than those of the ethereal matter, since the latter flows
in their interstices. It is true that there remains here some
difficulty in those experiments in which this interior reflexion
occurs without the particles of air being able to contribute to it, as
in vessels or tubes from which the air has been extracted.
Experience, moreover, teaches us that these two reflexions are of
nearly equal force, and that in different transparent bodies they are
so much the stronger as the refraction of these bodies is the greater.
Thus one sees manifestly that the reflexion of glass is stronger than
that of water, and that of diamond stronger than that of glass.
I will finish this theory of refraction by demonstrating a remarkable
proposition which depends on it; namely, that a ray of light in order
to go from one point to another, when these points are in different
media, is refracted in such wise at the plane surface which joins
these two media that it employs the least possible time: and exactly
the same happens in the case of reflexion against a plane surface. Mr.
Fermat was the first to propound this property of refraction, holding
with us, and directly counter to the opinion of Mr. Des Cartes, that
light passes more slowly through glass and water than through air.
But he assumed besides this a constant ratio of Sines, which we have
just proved by these different degrees of velocity alone: or rather,
what is equivalent, he assumed not only that the velocities were
different but that the light took the least time possible for its
passage, and thence deduced the constant ratio of the Sines. His
demonstration, which may be seen in his printed works, and in the
volume of letters of Mr. Des Cartes, is very long; wherefore I give
here another which is simpler and easier.
[Illustration]
Let KF be the plane surface; A the point in the medium which the light
traverses more easily, as the air; C the point in the other which is
more difficult to penetrate, as water. And suppose that a ray has come
from A, by B, to C, having been refracted at B according to the law
demonstrated a little before; that is to say that, having drawn PBQ,
which cuts the plane at right angles, let the sine of the angle ABP
have to the sine of the angle CBQ the same ratio as the velocity of
light in the medium where A is to the velocity of light in the medium
where C is. It is to be shown that the time of passage of light along
AB and BC taken together, is the shortest that can be. Let us assume
that it may have come by other lines, and, in the first place, along
AF, FC, so that the point of refraction F may be further from B than
the point A; and let AO be a line perpendicular to AB, and FO parallel
to AB; BH perpendicular to FO, and FG to BC.
Since then the angle HBF is equal to PBA, and the angle BFG equal to
QBC, it follows that the sine of the angle HBF will also have the same
ratio to the sine of BFG, as the velocity of light in the medium A is
to its velocity in the medium C. But these sines are the straight
lines HF, BG, if we take BF as the semi-diameter of a circle. Then
these lines HF, BG, will bear to one another the said ratio of the
velocities. And, therefore, the time of the light along HF, supposing
that the ray had been OF, would be equal to the time along BG in the
interior of the medium C. But the time along AB is equal to the time
along OH; therefore the time along OF is equal to the time along AB,
BG. Again the time along FC is greater than that along GC; then the
time along OFC will be longer than that along ABC. But AF is longer
than OF, then the time along AFC will by just so much more exceed the
time along ABC.
Now let us assume that the ray has come from A to C along AK, KC; the
point of refraction K being nearer to A than the point B is; and let
CN be the perpendicular upon BC, KN parallel to BC: BM perpendicular
upon KN, and KL upon BA.
Here BL and KM are the sines of angles BKL, KBM; that is to say, of
the angles PBA, QBC; and therefore they are to one another as the
velocity of light in the medium A is to the velocity in the medium C.
Then the time along LB is equal to the time along KM; and since the
time along BC is equal to the time along MN, the time along LBC will
be equal to the time along KMN. But the time along AK is longer than
that along AL: hence the time along AKN is longer than that along ABC.
And KC being longer than KN, the time along AKC will exceed, by as
much more, the time along ABC. Hence it appears that the time along
ABC is the shortest possible; which was to be proven.